How to separate in object mode?
You have made two different meshes but they are still parts of the same object. To make a new object, go back in Edit mode, select the separate mesh with L, then press P > Selection, this part will be a new object.

By the way, to create a separate object you don't have to separate the part of the mesh with Y or V, you can just select the part of the mesh you want to separate, then P (separate) > Selection.